DEFLECTION OF BEAMS AND SHAFTS

Today’s Objectives: Students will be able to:

a) Determine the deflection and slope at specific points on beams and shafts,

using various analytical methods including:

1. The integration method

2. The use of discontinuity functions

3. The method of superposition

In-class Activities:

• Reading Quiz

• Applications

• Elastic Curve

• Integration Method

• Use of discontinuity functions

• Method of superposition

• Moment area method

• Concept Quiz

APPLICATIONS

ELASTIC CURVE

- The deflection diagram of the longitudinal axis that passes through the

centroid of each cross-sectional area of the beam is called the elastic curve,

which is characterized by the deflection and slope along the curve. E.g.

Fig. 12-1

ELASTIC CURVE

- Moment-curvature relationship:

- Sign convention:

Fig. 12-2

Fig. 12-3 Fig. 12-4

ELASTIC CURVE

- Consider a segment of width dx, the strain in arc ds, located at a position y

from the neutral axis is ε = (ds’ – ds)/ds. However, ds = dx = ρdθ and ds’ = (ρ-

y) dθ, and so ε = [(ρ – y) dθ – ρdθ ] / (ρdθ), or

Fig. 12-5

1

ρ = – ε

y Comparing with the Hooke’s Law ε = σ / E and the flexure formula σ = -My/I

We have

1

ρ =

M

EI or

1

ρ = – σ

Ey

SLOPE AND DISPLACEMENT BY

INTEGATION Kinematic relationship between radius of curvature ρ and location x:

1

ρ = d2v / dx2

[ 1 + (dv/dx)2 ] 3/2

Then using the moment curvature equation, we have

M

EI = 1

ρ = d2v / dx2

[ 1 + (dv/dx)2 ] 3/2 ≈ d2v

dx2

Or M = EIv”

Since V = dM/dx, so V = EIv”’

Also –w = dV/dx, so –w = EIv””

The above three equations will be used to find the elastic curve by integration

Note: M is a function of x

Note: V is a function of x

Note: w is a function of x

SLOPE AND DISPLACEMENT BY

INTEGATION (CONT.)

Sign convention:

Fig. 12-8

SLOPE AND

DISPLACEMENT BY

INTEGATION (CONT.)

Boundary Conditions:

Table 12 -1

• The integration constants can be determined by

imposing the boundary conditions, or

• Continuity condition at specific locations

EXAMPLE

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE 2

EXAMPLE 2 (CONTINUED)

EXAMPLE 2 (CONTINUED)

EXAMPLE 2 (CONTINUED)

EXAMPLE 2 (CONTINUED)

USE OF DISCONTINOUS

FUNCTIONS Macaulay functions:

< x – a>n = { 0

(x – a )n

n ≥ 0

for x < a

for x ≥ a

Integration of Macaulay functions:

∫ <x – a>n dx = + C <x – a>n+1

n + 1

Table 12-2

USE OF DISCONTINOUS FUNCTIONS (CONT.)

Singularity Functions:

w = P <x – a>-1 = { 0 for x ≠ a

P for x = a

w = M0 <x – a>-2 = { 0 for x ≠ a

M0 for x = a

Fig. 12-16

Fig. 12-15

USE OF DISCONTINOUS FUNCTIONS (CONT.)

Fig. 12-17

Note: Integration of these two singularity functions yields results that are

different from those of Macaulay functions. Specifically,

∫ <x –a>n = <x – a>n+1 , n = -1, -2

Examples of how to use discontinuity functions to describe the loading or internal

moment in a beam:

w = - R1 <x – 0>-1 + P <x – a>-1 – M0<x – b>-2 + w0<x – c>0

M = R1 <x – 0> - P <x – a>) + M0<x – b>0 – (1/2)w0<x – c>2

EXAMPLE

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

METHOD OF SUPERPOSITION

• Necessary conditions to be satisfied: 1. The load w(x) is linearly related to the deflection v(x),

2. The load is assumed not to change significantly the original geometry of

the beam of shaft.

Then, it is possible to find the slope and displacement at a

point on a beam subjected to several different loadings by

algebraically adding the effects of its various component

parts.

EXAMPLE

EXAMPLE (CONTINUED)

EXAMPLE (CONTINUED)

CONCEPT QUIZ

1) The moment-curvature equation 1/ρ = M/EI is applicable to

A) Statically determined member only

B) Beams having uniform cross-sections only

C) Beams having constant Young’s Modulus E only

D) Beams having varying moment of inertia I.

2) The flexure equations imply that

A) Slope and deflection at a point of a beam are independent

B) Moment and shear at a point of a beam are independent

C) Maximum moment occurs at the locations where the shear is zero

D) Maximum moment occurs at the inflection point.

READING QUIZ

1) The slope angle θ in flexure equations is

A) Measured in degree C) Exactly equal to dv/dx

B) Measured in radian D) None of the above

2) The load must be limited to a magnitude so that not to change significantly

the original geometry of the beam. This is the assumption for

A) The method of superposition C) The method of integration

B) The moment area method D) All of them